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If you are new to the Windows Azure platform, you might want to understand about it in a staged and phased way.
This post gives you a roadmap of the content (written and video) that already exists on the Internet. You can go through the content, piece-by-piece and gradually understand the motivation for Azure, the architectural principles and eventually end up building and deploying your own application to the Windows Azure cloud.
If you are already experienced with Windows Azure, you might want to understand the new features which were announced at the Professional Developers Conference in November. This post will help you get up to speed.
Read the Wikipedia definition of Cloud Computing. There is no agreed definition but most of them incorporate these ideas and this is as good a description as any. Some will disagree, most will agree. This is a good place to get you started.
You can learn more at the Windows Azure platform portal.